arXiv:1405.1054v2 [hep-ph] 28 Dec 2015 · 2016-12-12 · Best ﬁt values of the oscillation pa-rameters.2 Parameter Bestﬁtvalue sin2 12 0.307 sin2 13 0.0241 sin2 ... Neutrino Experiment

Optimal configurations of the Deep Underground Neutrino Experiment


Vernon Barger∗

Department of Physics, University of Wisconsin, Madison, WI 53706, USA

Atri Bhattacharya†

Department of Physics, University of Arizona, Tucson, AZ 85721, USA

Animesh Chatterjee‡

University of Texas at Arlington, Arlington, TX 76019, USA

Raj Gandhi§

Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India

Danny Marfatia¶

Department of Physics and Astronomy, University of Hawaii at Manoa, Honolulu, HI 96822,

USA

Mehedi Masud‖

Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India

We perform a comprehensive study of the ability of the Deep Underground NeutrinoExperiment (DUNE) to answer outstanding questions in the neutrino sector. We considerthe sensitivities to the mass hierarchy, the octant of θ23 and to CP violation using datafrom beam and atmospheric neutrinos. We evaluate the dependencies on the precisionwith which θ13 will be measured by reactor experiments, on the detector size, beampower and exposure time, on detector magnetization, and on the systematic uncertaintiesachievable with and without a near detector. We find that a 35 kt far detector in DUNEwith a near detector will resolve the eight-fold degeneracy that is intrinsic to long baselineexperiments and will meet the primary goals of oscillation physics that it is designed for.

∗[email protected]†[email protected]‡[email protected]§[email protected]¶[email protected]‖[email protected]

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Table 1. Best fit values of the oscillation pa-rameters.2

Parameter Best fit valuesin2 θ12 0.307sin2 θ13 0.0241

sin2 θ23 (lower octant) 0.427sin2 θ23 (higher octant) 0.613

∆m221 7.54× 10−5 eV2

|∆m231| 2.43× 10−3 eV2

δCP 0

1. Introduction

Neutrino oscillations have by now been conclusively established by several pioneer-ing experiments. It is now understood that the mixing between the three neutrinoflavors is governed by the so-called PMNS mixing matrix,

UPMNS =

c12c13 s12c13 s13e−iδCP

−s12c23 − c12s13s23eiδCP c12c23 − s12s13s23e

iδCP c13s23

s12s23 − c12s13c23eiδCP −c12s23 − s12s13c23e

iδCP c13c23

, (1)

and the mass-squared differences: ∆m231 = m2

3 −m21 and ∆m2

21. Here, cij and sijare cos θij and sin θij respectively, for the three mixing angles θ12, θ23 and θ13, andδCP is a (Dirac) CP phase. While solar and atmospheric neutrino experiments havedetermined the first two mixing angles quite precisely, reactor experiments havemade remarkable progress in determining θ13.1 (See Table 1 for the values of theoscillation parameters used in our work.)

Now that θ13 has been conclusively shown to be non-zero and not too small,1,3

the focus of neutrino oscillation experiments has shifted to the measurement of δCPthat determines whether or not oscillating neutrinos violate CP. A second importantunanswered question for model building is whether the mass hierarchy is normalwith ∆m2

31 > 0, or inverted with ∆m231 < 0. Finally, the question of whether θ23 is

larger or smaller than π/4 bears on models based on lepton symmetries.An effort towards resolving the above issues and thereby taking us a step closer

to completing our knowledge of the neutrino mass matrix, is the Deep UndergroundNeutrino Experiment (DUNE).a DUNE will employ a large liquid argon far detector(FD). It is expected to be placed underground in the Homestake mine at a distanceof 1300 km from Fermilab, from which a neutrino beam will be directed towards thedetector. Large-mass Liquid Argon Time Projection Chambers (LAr-TPCs) haveunprecedented capabilities for the detection of neutrino interactions due to preciseand sensitive spatial and calorimetric resolution. However, the final configuration

aThe inputs we use, and the corresponding references, pertain to the erstwhile Long BaselineNeutrino Experiment (LBNE),4,5 which has undergone a new phase of internationalisation andexpansion. This has led to a change in the name of the experiment, to DUNE. Nonetheless, it isexpected that the configuration we assume here vis a vis fluxes, baseline and energies will remainlargely intact.


3

of the experiment is still under discussion.6,7 The sensitivity of DUNE to the masshierarchy, to CP violation and to the octant of θ23 depends on, among other things,how well other oscillation parameters are known, on the amount of data that can betaken in a reasonable time frame, on the systematic uncertainties that compromisethe data, and on the charge discrimination capability of the detector. Our goal (inthis extension of our previous work7) is to study how these factors affect DUNEin various experimental configurations. Other recent studies of some of the physicscapabilities of DUNE can be found in Ref.8 .

1.1. Objectives

The various considerations of our work are motivated by possible configurationsfor DUNE in the initial phase of its program. The initial stage of DUNE will, atthe very least, permit the construction of an unmagnetized 10 kt FD underground.Several improvements upon this basic configuration are under consideration. Thesemight include

• upgrading the FD to 35 kt for improved statistics,• having a precision near detector (ND) for better calibration of the initial

flux and reducing the involved systematic uncertainties,• magnetizing the FD to make it possible to distinguish between particles

and antiparticles in the atmospheric neutrino flux.b

It must be noted that some of the above upgrades would also have supplementarybenefits — an ND, for example, will also allow precision studies of the involvedneutrino nucleon cross sections, thereby reducing present uncertainties.

Since it might not be feasible to combine all of the above upgrades into an initialDUNE configuration, we evaluate which combination would be most beneficial asfar as the physics of neutrino oscillations is concerned. Specifically, we study thefollowing experimental configurations:

(1) A beam experiment with and without an ND.(2) An atmospheric neutrino experiment.(3) An experiment with and without an ND that combines beam and atmo-

spheric neutrino data collected at the FD.(4) A global configuration that combines DUNE data (with and without ND)

with NOνA9 and T2K10 data.

Our study will highlight the benefits of

(1) building a larger 35 kt FD as opposed to a 10 kt detector,(2) higher exposure (kt-MW-yr),11

bThe beam experiment would have the neutrino and antineutrino runs happen asynchronously.Thus, the events from the two would be naturally distinguished and magnetization of the FDwould not affect its results.


4

Table 2. Systematic uncertainties for signal andbackground channels in DUNE.5,15

Detector configuration SystematicsSignal Background

With ND νe: 1% νe: 1%νµ: 1% νµ: 5%

Without ND νe: 5% νe: 10%νµ: 5% νµ: 45%

(3) magnetizing the FD versus having an unmagnetized detector volume,(4) utilizing atmospheric neutrinos,(5) precision θ13 measurements,

Throughout, we estimate how data from NOνA and T2K improve DUNE sensitivi-ties; for previous discussions see Refs.4,12 . All simulations for the beam experimentshave been done numerically with the GLoBES software.13

2. Experimental specifications and analysis methodology

We consider neutrinos resulting from a 120 GeV proton beam from Fermilab with abeam power of 700 kW and an uptime of 1.65×107 seconds per year (or equivalently6 × 1020 protons on target (POT) per year) incident at a LAr FD at a baseline of1300 km; an upgrade to a 2.3 MW beam is a possibility. As noted in Ref.14 , thephysics performance is roughly equivalent for proton beam energies that exceed60 GeV. We simulate events at the FD using the GLoBES software13 and the fluxesused by the DUNE collaboration.

If the FD is placed underground, it is also sensitive to atmospheric neutrinos.We simulate atmospheric neutrino events (as described in Appendix 7) and both νeappearance and νµ disappearance channel events from the beam in the neutrino andantineutrino modes with an event reconstruction efficiency of 85%. In our simulationof the DUNE beam experiment we employ the signal and background systematicsfor νe appearance and νµ disappearance channels from Refs.5,15 ; see Table 2.c Forthe energy resolutions, we have used the method of bin-based automatic energysmearing with σ

E = 0.20√E

for νµ events and σE = 0.15

E for νe events; see the appendixof Ref.5 An alternative is to use migration matrices.5,12 For NOνA9 and T2K,10

the relevant parameters are given in Tables 3 and 4, respectively. An up-to-datedescription of the NOνA and T2K experiments can be found in Ref.12 .

cOur ND analysis represents the most obvious benefit that the beam experiment will reap with anND, viz., improvement in systematics for the signal and background events. In addition, an ND willalso improve our understanding of the fluxes and cross sections, thereby reducing shape-relateduncertainties in the analysis. We do not attempt an exploration of this facet of the ND becausethe exact nature of the improvement would depend to a large extent on the specifics of the ND,which for the DUNE is yet in the planning stage. Our ND analysis represents a worst-case scenariofor improvement in the systematics.


5

Table 3. Systematic uncertainties for NOνA.9


15 kt TASD νe: 5% νe: 10%3 yrs. ν + 3 yrs. ν6× 1020 POT/yr νµ: 2% νµ: 10%

with a 700 kW beam

Table 4. Systematic uncertainties for T2K.10


22.5 kt water Cherenkov νe: 5% νe: 5%5 yrs. ν

8.3× 1020 POT/yr νµ: 5% νµ: 5%with a 770 kW beam

Throughout, we assume that placing the detector underground does not resultin a significant change of the signal and background analysis, apart from makingthe detector also sensitive to the atmospheric neutrino events, and thus allowinga combined analysis of the beam and atmospheric data over the duration of theexperiment.

We assume that the beam is run in the neutrino mode for a period of five years,and thereafter in the antineutrino mode for five more years.

For the atmospheric neutrino analysis, and consequently the combined beam andatmospheric analysis, it becomes important to consider both a magnetized and anunmagnetized LAr detector. In the former case, the detector sensitivity, especiallyfor the resolution of the mass hierarchy, is significantly improved over the latter,due to its ability to distinguish between particles and antiparticles. This, however,is only partly applicable to the νe events, because for a 10 kt volume detector, it isdifficult to distinguish between the tracks arising from of νe and νe interactions. Thisdifficulty arises because pair-production and bremsstrahlung sets in with increasingenergies, and above ≈ 5 GeV, the detector completely loses its ability to distinguishbetween these CP conjugate pairs. On the other hand, due to their tracks beingsignificantly longer, νµ and νµ events are clearly distinguishable at all accessibleenergies. We implement this in our detector simulation for the atmospheric neutrinoand combined analysis.

For our simulation of atmospheric neutrino data, the energy and angular reso-lutions of the detector are as in Table 5.16 The atmospheric fluxes are taken fromRef.17 , the flux and systematic uncertainties from Ref.18 , and the density profileof the earth from Ref.19 .

The charge identification capability of the detector is incorporated as discussedin Ref.16 . For electron events, we conservatively assume a 20% probability of chargeidentification in the energy range 1 − 5 GeV, and no capability for events withenergies above 5 GeV. Since the muon charge identification capability of a LAr-


6

Table 5. Detector parameters used for the analysis of atmosphericneutrinos.16

Rapidity (y) 0.45 for ν0.30 for ν

Energy Resolution (∆E)√

(0.01)2 + (0.15)2/(yEν) + (0.03)2

Angular Resolution (∆θ) 3.2 for νµ2.8 for νe

Detector efficiency (E) 85%

TPC is excellent for energies between 1 and 10 GeV, we have assumed it to be100%.

For the combined analysis of beam and atmospheric events, we first calculatethe χ2 separately from the atmospheric analysis (using our code) and the DUNEbeam analysis (using GLoBES) for a set of fixed oscillation parameters. After addingthese two fixed parameter χ2 values, we marginalize over θ13, θ23, |∆m2

31| and δCP

to get the minimized χ2; see Appendix 7. The procedure is similar for our combinedanalysis of DUNE, T2K and NOνA data.

3. Mass hierarchy

Since δCP will likely remain undetermined by experiments preceeding DUNE, weanalyze the sensitivity to the mass hierarchy as a function of this parameter. Theanalysis is carried out by assuming one of the hierarchies to be true and thendetermining by means of a χ2 test, how well the other hierarchy can be excluded.We marginalize over the present day uncertainties of each of the prior determinedparameters.

The δCP dependence of the sensitivity to the mass hierarchy arises through theoscillation probability,20

Pµe = T 21 + T 2

2 + 2T1T2 cos(δCP + ∆) , (2)

where,

T1 = α sin 2θ12 cos θ23sin(x∆)

x (3)

T2 = sin 2θ13 sin θ23sin[(1−x)∆]

(1−x) , (4)

and α =∆m2

21

∆m231, x = 2EV

∆m231, ∆ =

∆m231L

4E , and V = ±2√

2GFNe is the matterpotential (positive for neutrinos and negative for antineutrinos). The only otherrelevant probability, Pµµ, is mildly dependent on δCP.

3.1. Analysis with a 35 kt unmagnetized LAr FD

As is evident from Fig. 1, mass hierarchy resolution benefits significantly from havingan ND. But, note that the results with or without an ND are similar for regions of


7

3σ5σ

NH

Δχ2

0

50

100

150

200

250

300

δCP / π−1 −0.5 0 0.5 1

IH

δCP / π−1 −0.5 0 0.5 1

Solid curves : with ND Broken curves : w/o NDAtmospheric onlyBeam onlyAtm & BeamAtm + NOvA + T2K + LBNE

Fig. 1. Sensitivity to the mass hierarchy as a function of true δCP for a true normal hierarchy(NH) and a true inverted hierarchy (IH) with an 350 kt-yr exposure at the unmagnetized fardetector configured with and without a near detector (ND). A run-time of 5 years each (3× 1021

protons on target) with a ν and ν beam is assumed. The combined sensitivity with NOνA (15 ktTASD, 3 yrs. ν + 3 yrs. ν) and T2K (22.5 kt water cerenkov, 5 yrs. ν) data is also shown.

5σ

3σ

NH

Δχ2

0

20

40

60

80

100

δCP / π−1 −0.5 0 0.5 1

IH

δCP / π−1 −0.5 0 0.5 1

Solid curves : with ND Broken curves : w/o ND

Atmospheric onlyBeam onlyAtm & BeamAtm + NOvA + T2K + LBNE

Fig. 2. Similar to Fig. 1 but for a 100 kt-yr unmagnetized LAr FD.

δCP where the sensitivity is worse (δCP ∈ [45, 135] for the normal hierarchy andδCP ∈ [−135,−45] for the inverted hierarchy).

Since the wrong hierarchy can be excluded by the DUNE beam-only experimentto significantly more than 5σ with an unmagnetized LAr FD and an exposure of 350kt-yrs. without the help of ND, the added contributions of both the atmosphericneutrinos, and the better signal and background systematics provided by an ND,are not essential for this measurement.

Similar conclusions vis-a-vis the near detector can be drawn for a 10 kt FD from


8

NHf (σ

> 3

)

0

0.2

0.4

0.6

0.8

1

Exposure [kt-MW-yr]

0 10 20 30 40 50 60

IH

Exposure [kt-MW-yr]

10 20 30 40 50 60

Beam onlyAtm & BeamAtm + NOvA +T2K + LBNE

Solid curves: with NDBroken curves: w/o ND

Fig. 3. The fraction of CP phases for which the sensitivity to the mass hierarchy exceeds 3σas a function of DUNE exposure, for different unmagnetized detector configurations. The timeexposure refers to calendar years for DUNE with 1.65×107 seconds of uptime per year. The entireNOνA and T2K datasets are assumed to be available when DUNE starts taking data (and do notcontribute to the exposure shown).

Fig. 2. For a 100 kt-yr exposure, the combined analysis resolves the hierarchy tomore than 5σ for a large δCP fraction, and to more than 3σ for all values of δCP.

3.2. Exposure analysis

We now evaluate the exposure needed to resolve the mass hierarchy for the entirerange of δCP. In Fig. 3, we show the CP fraction (f(σ > 3)) for which the sensitivityto mass hierarchy is greater than 3σ, as a function of exposure. Salient points evidentfrom Fig. 3 are:

• For a beam only analysis, we see that a 3σ determination of the hierarchyfor any δCP value is possible with a roughly 50 kt-MW-yr exposure. Thismeans the hierarchy can be resolved by a 35 kt FD and a 700 kW beam intwo years.• A near detector does not reduce the exposure needed for a 3σ measurement.• Information from atmospheric neutrinos reduces the exposure required to

about 45 kt-MW-yr.• A further combination with NOνA and T2K data provides minor improve-

ment.

3.3. Variation of systematics

In Fig. 4, we show the maximum sensitivity to the mass hierarchy for the entireδCP (true) space as a function of the exposure. We have allowed for variations insystematics for DUNE with an ND that are 3 times as large or small as those inTable 2). The width of the band produced by this procedure may be considered


9

σ =

√(Δ

χ2 )

0

2

4

6

8

10

12

Exposure [kt-MW-yr]100 200 300 400 500

NH

5σ

3σ

Exposure [kt-MW-yr]100 200 300 400 500

IH

Fig. 4. Maximum sensitivity to the mass hierarchy for all values of δCP allowing for differentsystematics (see Sec. 3.3), as a function of exposure. Only beam data (with both an FD and ND)have been considered.

as a measure of the effect of systematics on the hierarchy sensitivity when an NDis used (which seems to be the likely scenario in practice) along with an FD. Weobserve the following features from Fig. 4:

• For the NH (left panel), both 3σ and 5σ levels of sensitivity can be reachedwith an exposure of about 50 and 120 kt-MW-yr, respectively. This is con-sistent with Fig. 3 wherein the solid magenta curve in the left panel reachesunity at roughly 50 kt-MW-yr.• For exposures below ∼ 20 kt-MW-yr, the sensitivity is not statistically

significant (. 1.5σ).• The variation of systematics has a small effect on the sensitivities for lower

exposures (. 100 kt-MW-yr) and the effect gets slightly magnified for largerexposures, as evident from the widening of the bands.• The hierarchy sensitivity for a true IH scenario (right panel of Fig. 4) shows

qualitatively similar behaviour as that for NH.

3.4. Effect of magnetization

In Fig. 5, we compare the sensitivity to the mass hierarchy of an unmagnetized andmagnetized 100 kt-yr LAr FD for a true NH. As discussed earlier, magnetizing thedetector volume holds significance for the atmospheric neutrino analysis, since itallows the discrimination of neutrinos and antineutrinos in the flux. Consequently,for the magnetized detector the atmospheric neutrinos alone contribute an almost3σ sensitivity, thus also enhancing the combined sensitivity; the beam-only resultsremain unaffected by magnetization.


10

5σ

3σ

Unmag

Δχ2

0

20

40

60

80

100

δCP / π−1 −0.5 0 0.5 1

Mag

δCP / π−1 −0.5 0 0.5 1

Atmospheric onlyBeam onlyAtm & Beam

Solid curves: with ND Broken curves: w/o ND

Fig. 5. Sensitivity to the mass hierarchy with a 100 kt-yr exposure with a magnetized (mag) andunmagnetized (unmag) FD. The true hierarchy is normal.

3.5. Some remarks on the results

Some understanding of the qualitative nature of the results may be gleaned fromconsidering the relevant expressions at the level of oscillation probabilities.

Ignoring systematic uncertainties, a theoretical event-rate dataset (N th) and anexperimental one (N exp) can be used to define

χ2 ∼∑i

(N th(i)−N exp(i))2

N exp(i). (5)

Since the event rate at energy E in the e− appearance channel is given by Pµe(E)×Φ(E)× σCC(E), the sensitivity to the mass hierarchy follows the behavior of

ΠMH(A,B) =(PAµe − PB

µe

)2, (6)

where A represents the assumed true hierarchy and B represents the test hierarchy.The opposing natures of the sensitivities seen for the NH as true and IH as truescenarios respectively can be related to the δCP-dependent phase difference betweenΠMH(NH, IH) and ΠMH(IH,NH) at energies where the flux is high, i.e., 1.5–3.5 GeV;see Fig. 16 in Appendix B. The oscillatory nature of ∆χ2 with respect to δCP ineach case can be traced back to Eq. (6) too.

Because of the strong parameter space degeneracies involved in the appearancechannel probability [Eq. (2)], neither the T2K nor the NOνA experiments are ca-pable of significantly improving the mass hierarchy sensitivities in their respectiveconfigurations. It is apparent that the mass hierarchy study benefits immenselyfrom the longer baseline as well as improved systematics of the DUNE set-up ascompared to T2K and NOνA.

Since the mass hierarchy will be determined at 3σ with relatively little exposure(see Fig. 3), henceforth, we assume the mass hierarchy to be known. It is well known


11

that studies of the octant degeneracy and CP violation benefit significantly fromknowledge of the mass hierarchy.

4. Octant degeneracy

We test the sensitivity to the θ23 octant by using the true value to be equivalentto the present best-fits sin2(θtrue

23 ) = 0.427 (0.613) for the lower (higher) octants(LO/HO), except for Fig. 7, where we show the sensitivities for a range of θ23 valuesin both octants. The ∆χ2 in each case represents the sensitivity to disfavoring theopposite octant when a particular choice of θtrue

23 is made.The results for the octant analysis have one important feature — the inverted

hierarchy scenario shows almost no variation in sensitivity with δCP. This followsfrom the well-known result that when neutrino masses are arranged in an invertedhierarchy, the contribution to the ∆χ2 comes almost equally from antineutrinosand neutrinos, but with nearly opposite values of δCP, while if they conform to thenormal hierarchy, the neutrinos dominate over the antineutrinos and the overallresult largely traces the features of the neutrino-only ∆χ2.


Figure 6 shows the sensitivity to the octant of θ23 for a given mass hierarchy.d Byand large, a beam only analysis with an ND provides better sensitivity than thecombined analysis with atmospheric data without an ND. Only for the lower octantand normal hierarchy (LO-NH), do the sensitivities almost coincide.

Figure 7 shows the octant sensitivity as a function of true θ23. No sensitivityis expected for θ23 = 45. The sensitivity with an ND is slightly greater than thecombined analysis with atmospheric data without an ND if the true hierarchy isinverted, and the converse is true for a normal hierarchy.

4.2. Effect of θ13 precision

Resolving the octant degeneracy depends greatly on the precision with which θ13

is known. As Fig. 8 shows, a 2% uncertainty on sin2 2θ13 significantly improves theoctant sensitivity compared to a 5% uncertainty on sin2 2θ13.


Results of our exposure analysis are shown separately for the two octants in Figs. 9and 10. We note the following:

• In the case of LO-NH, for exposures below 50 kt-MW-yr, f(σ > 3) risesslowly with exposure for a beam-only experiment with or without a near

dThe χ2 analysis converges to the minimum extremely slowly in the HO case of the combinedsetup. Hence, for results in this section, we only show representative plots for the combined setupfor the LO case. We expect qualitatively similar results for the HO case.


12

3σ

5σ

LO-NH

Δχ2

0

20

40

60

80LO-IH

Solid curves : with ND Broken curves : w/o NDAtmospheric onlyBeam onlyAtm & BeamAtm + NOvA + T2K + LBNE

3σ

5σ

HO-NH

Δχ2

0

20

40

60

80

δCP/π−1 −0.5 0 0.5 1

HO-IH

δCP/π−1 −0.5 0 0.5 1

Fig. 6. Sensitivity to the octant of θ23 with σ(sin2 2θ13) = 0.05× sin2 2θ13 and sin2 θ23 = 0.427

in the lower octant (LO) and sin2 θ23 = 0.613 in the higher octant (HO), for both hierarchies and a350 kt-yr unmagnetized FD exposure configured with and without an ND. Representative resultsof the combined sensitivity with atmospheric data, and with NOνA and T2K are shown for thelower octant.

detector; see Fig. 9. Above this exposure, the curves steepen and eventuallythe degeneracy is broken for all δCP values for a 75 kt-MW-yr exposure.Thus, the octant will be known at 3σ in one year after the mass hierarchyis determined with a 35 kt detector and 700 kW beam.• The right panels of Figs. 9 and 10 show the IH case. The steepness of

the curves can be understood from Fig. 6 which shows that ∆χ2 is almostindependent of δCP for the IH. As the exposure is increased, ∆χ2 increases,and above a critical exposure, the octant degeneracy is broken at 3σ foralmost all values of δCP with a small increment in exposure.• From Fig. 10, we observe that the lower octant can be ruled out at 3σ for

the entire δCP parameter space with less than a 40 kt-MW-yr exposure,with or without a near detector.


In Fig. 11, we plot the maximum sensitivity to the octant that can be achievedfor all values of δCP. We express the sensitivity as a band, obtained by varying the


13

NH

3σ

5σ

Δχ2

25

50

75

100

125

150

True θ23 [Deg.]36 38 40 42 44 46 48 50 52 54

IH

True θ23 [Deg.]36 38 40 42 44 46 48 50 52 54

Atmospheric onlyBeam only

Atm & BeamAtm + NOvA + T2K + LBNE

Solid curves : with ND Broken curves: w/o ND

Fig. 7. Sensitivity to the octant for a 350 kt-yr unmagnetized FD exposure as a function of θ23and with σ(sin2 2θ13) = 0.05× sin2 2θ13.

3σ

LO-NH

5σ

Δχ2

0

20

40

60

80

100

LO-IH

Atmospheric onlyBeam only

Atm & BeamAtm + NOvA + T2K + LBNE

Thin curves: σ(sin22θ13) = 0.02 × sin22θ13 Thick curves: σ(sin22θ13) = 0.05 × sin22θ13

HO-NH

3σ

5σ

Δχ2

0

20

40

60

80

100

δCP/ π−1 −0.5 0 0.5 1

HO-IH

δCP/ π−1 −0.5 0 0.5 1

Fig. 8. Octant sensitivity for σ(sin2 2θ13) = 0.02× sin2 2θ13 and σ(sin2 2θ13) = 0.05× sin2 2θ13for a 350 kt-yr unmagnetized FD and an ND. Representative results of the combined sensitivitywith atmospheric data, and with NOνA and T2K are shown for the lower octant.a


14

LO-NH

f (σ

> 3)

0

0.2

0.4

0.6

0.8

1

Exposure [kt-MW-yr]40 50 60 70 80 90 100

LO-IH

Exposure [kt-MW-yr]50 60 70 80 90 100

Beam onlyAtm & BeamAtm + NOvA + T2K + LBNE


Fig. 9. The fraction of CP phases for which the sensitivity to the octant exceeds 3σ as a functionof exposure, for θ23 in the lower octant (LO) and σ(sin2 2θ13) = 0.05× sin2 2θ13.

HO-NH

f (σ

> 3)

0

0.2

0.4

0.6

0.8

1

Exposure [kt-MW-yr]20 25 30 35 40

HO-IH

Exposure [kt-MW-yr]25 30 35 40

Beam onlyAtm & BeamAtm + NOvA + T2K + LBNE


Fig. 10. The fraction of CP phases for which the sensitivity to the octant exceeds 3σ as a functionof exposure, for θ23 in the higher octant (HO) and σ(sin2 2θ13) = 0.05× sin2 2θ13.

systematics as described in Sec. 3.3). We note from Fig. 11 that,

• The 3σ sensitivity level can be achieved with ∼ 70 kt-MW-yr for bothhierarchies. This is consistent with Fig. 9.• For the NH, it takes less (∼ 400 kt-MW-yr) exposure to obtain 5σ sensitivity

compared to the true IH scenario (∼ 500 kt-MW-yr).• The variation of systematics has a negligible effect on the sensitivity for

exposures . 70 kt-MW-yr until 3σ sensitivity is reached. Thereafter, toachieve the 5σ level, the sensitivities get slightly more affected by system-atics.


15

LO-NH

3σ

5σ

σ =

√(Δ

χ2 )

0

1

2

3

4

5

6

Exposure [kt-MW-yr.]100 200 300 400 500

LO-IH

Exposure [kt-MW-yr.]100 200 300 400 500

Fig. 11. Similar to Fig. 4, but for the maximum sensitivity to a true lower octant (LO).

NH

Solid curves: Mag Broken curves: Unmag

Atmospheric onlyAtm & Beam

3σ

5σ

Δχ2

0

10

20

30

40

50

60

True θ23 [Deg.]36 38 40 42 44 46 48 50 52 54

Fig. 12. Octant sensitivity of a magnetized and an unmagnetized detector with a 100 kt-yrexposure. The true hierarchy is assumed to be normal.


We see from Fig. 12, that magnetizing the detector has almost no effect on thesensitivity to the octant. From a practical standpoint, it is unlikely that a 35 kt FDwould be magnetized, with no consequence for octant sensitivity.

It is apparent that the octant resolution benefits significantly from the presence ofa calibrating ND. As Figs. 9 and 10 show, the consequent improvement in statisticsreduces the runtime required for the achievement of a 3σ significant resolution by


16

at least 8 yrs (for NH, both octants) and as much as 20 yrs for the LO-IH scenario.In contrast, the benefit of adding the atmospheric flux is modest, except in the

case of LO-NH where a combination of the atmospheric and beam sensitivities candrive the resolution to almost 5σ significance despite the absence of the ND. Whilethe atmospheric contribution is also large in the case of HO-NH, in this case thebeam-only epxeriment is already capable of resolving the degeneracy to more thana 3σ level by itself even without the ND. In the latter case, therefore, the additionalexpenditure that would be inevitably involved in building the FD underground,would not be justifiable.

The octant degeneracy resolution also greatly benefit from investment in in-creasing the FD volume to 35 kt from 10 kt, as is evident from the Figs. 9 and10.

5. CP violation

Of the six oscillation parameters, δCP is the least well known. Part of the reason forthis was the difficulty in experimentally determining the value of θ13. With reactorexperiments over the last three years having made significant progress toward theprecision determination of the latter, and it being established by now that the valueof θ13 is non-zero by a fair amount, the precision determination of δCP in a futureexperiment should be possible.

In the following we study the sensitivity of the DUNE to CP-violation in theneutrino sector brought about by a non-zero δCP phase. To determine the ∆χ2

that represents the experiment’s sensitivity to CP-violaion, we assume a test δCP

value of 0 (or π) and compute the ∆χ2 for any non-zero (or 6= π) true δCP. Sincethe disappearance channel probability Pµµ is only mildly sensitive to the δCP, CP-violation in the neutrino sector can only be studied by experiments sensitive to theappearance channel νµ → νe. It is obvious, given the nature of the latter channel’sprobability Pµe, that the maximum sensitivity will be due to values of true δCP

close to odd multiples of π/2.Due to the non-zero value of θ13 being now established, other experiments sen-

sitive to the νµ → νe appearance channel, including the T2K and NOνA, are alsostrongly poised to look for CP-violation. Consequently, this is one study wherecombining data from DUNE, T2K and NOνA proves to be significantly beneficial.


To study CP violation, reduced systematics courtesy the placement of an ND provesto be beneficial (Fig. 13). Maximal CP violation can be ruled out at more than5σ by a beam only analysis with a 350 kt-yr exposure in conjunction with theND. However, 5σ resolution toward ruling out maximal CP-violation can even beachieved despite the absence of an ND by combining results from the T2K, NOνAand the DUNE.


17

5σ

3σ

NH

Δχ2

0

20

40

60

80

δCP / π−1 −0.5 0 0.5 1

IH

δCP / π−1 −0.5 0 0.5 1

Solid curves : with ND Broken curves : w/o NDAtmospheric onlyBeam only

Atm & BeamAtm + NOvA +T2K + LBNE

Fig. 13. Sensitivity to CP violation for a 350 kt-yr unmagnetized FD exposure assumingσ(sin2 2θ13) = 0.05× sin2 2θ13.

NH

f (σ

> 3)

0

0.2

0.4

0.6

0.8

1

Exposure [kt-MW-yr]0 50 100 150 200 250 300 350

IH

Exposure [kt-MW-yr]50 100 150 200 250 300 350

Solid curves : with ND Broken curves : w/o NDBeam onlyAtm & BeamAtm + NOvA + T2K + LBNE

Fig. 14. The fraction of CP phases for which the sensitivity to CPV exceeds 3σ as a function ofexposure.


In Fig. 14, we show the CP fraction for which CP violation can be established at3σ. Needless to say, the CP fraction has to be less than unity since even an almostideal experiment cannot exclude CP violating values of the phase that are close tothe CP conserving values, 0 and π. In the context of CP violation, the CP fractionis a measure of how well an experiment can probe small CP violating effects. FromFig. 14, we find:

• There is no sensitivity to CP violation at the 3σ level for exposures smaller


18

σ =

√(Δ

χ2 )

0

1

2

3

4

5

Exposure [kt-MW-yr]200 400 600 800 1000

3σ

NH IH

Exposure [kt-MW-yr]200 400 600 800 1000

IH

Fig. 15. Similar to Figs. 4 and 11, but for the maximum sensitivity to CP violation for 70% ofthe δCP parameter space.

than about 35 kt-MW-yr. The sensitivity gradually increases with exposureand the CP fraction for which 3σ sensitivity is achieved approaches 0.4(without an ND) and 0.5 (with an ND) for a 125 kt-MW-yr exposure. TheCP fraction plateaus to a value below 0.8 for an exposure of 350 kt-MW-yrwith all data combined.• A near detector certainly improves the sensitivity to CP violation.


Figure 15 shows the maximum sensitivity to CP violation that can be achieved for70% of the δCP parameter space. As in Figs. 4 and 11, we show the sensitivity as aband on varying the systematics. The notable features of Fig. 15 are,

• To resolve CP violation at the level of 3σ for 70% region of the δCP space,a fairly long exposure is needed. For NH, it is roughly 400−500 kt-MW-yr.while for IH it is 350− 450 kt-MW-yr. depending on the systematics.)• For such long exposures, the sensitivity band becomes appreciably wide

indicating a strong dependence on the systematics. In comparison, the sen-sitivities to the mass hierarchy and octant were less dependent on system-atics since the corresponding exposures were smaller. This reinforces theneed for an ND.


As can be seen from Fig. 13, the sensitivity of atmospheric neutrinos to CPV isnegligible, hence magnetizing the detector does not help.


19

It is obvious that CP-violation is the study that stands to benefit most from thecombination of results from the T2K, NOνA and the DUNE. Even potentially lowsensitivity to maximal CP-violation due to the absence of ND can be overcome bythe combination of χ2 data from the three epxeriments. However, a large volume FD(35 kt) for the DUNE is almost certainly an absolute necessity, if any sensitivity toCP-violation has to be detected within a reasonable time frame, irrespective of thebenefits of combining results from other experiments such as the T2K and NOνA.

The atmospheric neutrino flux has no role to play in the resolution of this phys-ical problem.

6. Summary

We considered the Deep Underground Neutrino Experiment as either a 10 kt or35 kt LAr detector situated underground at the Homestake mine and taking datain a high intensity neutrino beam for 5 years and in an antineutrino beam foranother 5 years. For the 35 kt detector, we find that reduced systematic uncertaintiesafforded by a near detector greatly benefit the sensitivity to CP violation. However,a near detector provides only modest help with the octant degeneracy and is notnecessary for the determination of the mass hierarchy since the sensitivity withouta near detector is well above 5σ. Since magnetization is not currently feasible fora 35 kt detector, we only considered this possibility for a 10 kt detector. Whilethe sensitivity to the mass hierarchy from atmospheric neutrinos gets enhancedto almost 3σ, the combined beam and atmospheric data is not much affected bymagnetization. Also, magnetizing the detector does not help improve the sensitivityto the octant or to CP violation.

One thing is clear. A 35 kt DUNE will break all remaining vestiges of the eight-fold degeneracy that plagues long-baseline beam experiments20 and will answer allthe questions it is designed to address.

Acknowledgments

AB, AC, RG and MM thank Pomita Ghoshal and Sanjib Mishra for discussions. Thiswork was supported by US DOE grants DE-FG02-95ER40896 and DE-SC0010504.RG acknowledges the support of the XI Plan Neutrino Project under DAE.

Appendix

7. Atmospheric neutrino analysis

The simulation of atmospheric neutrino events and the subsequent χ2 analysis iscarried out by means of a C++ program. Our method is described below.

7.1. Event simulation

The total number of CC events is obtained by folding the relevant incident neutrinofluxes with the appropriate disappearance and appearance probabilities, relevant


20

CC cross sections, and the detector efficiency, resolution, mass, and exposure time.For our analysis, we consider neutrinos with energy in the range 1−10 GeV in 10uniform bins, and the cosine of the zenith angle θ in the range −1.0 to −0.1 in 18bins. The µ− event rate in an energy bin of width dE and in a solid angle bin ofwidth dΩ is,

d2Nµ

dΩ dE=

1

2π

[(d2Φµ

d cos θ dE

)Pµµ +

(d2Φe

d cos θ dE

)Peµ

]σCCDeff . (7)

Here Φµ and Φe are the νµ and νe atmospheric fluxes, Pµµ and Peµ are dis-appearance and appearance probabilities in obvious notation, σCC is the total CCcross section and Deff is the detector efficiency. The µ+ event rate is similar to theabove expression with the fluxes, probabilities and cross sections replaced by thosefor antimuons. Similarly, the e− event rate in a specific energy and zenith angle binis

d2Ne

dΩ dE=

1

2π

[(d2Φµ

d cos θ dE

)Pµe +

(d2Φe

d cos θ dE

)Pee

]σCCDeff , (8)

with the e+ event rate being expressed in terms of antineutrino fluxes, proba-bilities and cross sections.

We take into account the smearing in both energy and zenith angle, assuming aGaussian form for the resolution function, R. For energy, we use,

RE(Et,Em) =1√2πσ

exp

[− (Em − Et)

2

2σ2

]. (9)

Here, Em and Et denote the measured and true values of energy respectively. Thesmearing width σ is a function of Et.

The smearing function for the zenith angle is a bit more complicated becausethe direction of the incident neutrino is specified by two variables: the polar angleθt and the azimuthal angle φt. We denote both these angles together by Ωt. Themeasured direction of the neutrino, with polar angle θm and azimuthal angle φm,which together we denote by Ωm, is expected to be within a cone of half angle ∆θ ofthe true direction. The angular smearing is done in a small cone whose axis is givenby the direction θt, φt. The set of directions within the cone have different polarangles and azimuthal angles. Therefore, we need to construct a smearing functionwhich takes into account the change in the azimuthal coordinates as well. Such anangular smearing function is given by,

Rθ(Ωt,Ωm) = N exp

[− (θt − θm)2 + sin2 θt (φt − φm)2

2(∆θ)2

], (10)

where N is a normalisation constant.Now, the νµ event rate with the smearing factors taken into account is given by,

d2Nµ

dΩm dEm=

1

2π

∫ ∫dEt dΩt REN(Et,Em) Rθ(Ωt,Ωm)

[Φdµ Pµµ + Φd

e Peµ

]σCCDeff ,

(11)


21

and similarly for the νe event rate. We have introduced the notation,

(d2Φ/d cos θ dE)µ,e ≡ Φdµ,e.

Since REN(Et,Em) and Rθ(Ωt,Ωm) are Gaussian, they can easily be integratedover the true angle Ωt and the true energy Et. Then, integration over the measuredenergy Em and measured angle Ωm is carried out using the VEGAS Monte CarloAlgorithm.

7.2. χ2 analysis

The computation of χ2 for a fixed set of parameters is performed using the methodof pulls. This method allows us to take into account the various statistical andsystematic uncertainties in a straightforward way. The flux, cross sections and othersystematic uncertainties are included by allowing these inputs to deviate from theirstandard values in the computation of the expected rate in the i-jth bin, Nth

ij . Letthe kth input deviate from its standard value by σk ξk, where σk is its uncertainty.Then the value of Nth

ij with the modified inputs is

Nthij = Nth

ij (std) +

npull∑k=1

ckij ξk , (12)

where Nthij (std) is the expected rate in the i-jth bin calculated with the standard

values of the inputs and npull is the number of sources of uncertainty, which is 5 inour case. The ξk’s are called the pull variables and they determine the number ofσ’s by which the kth input deviates from its standard value. In Eq. (12), ck

ij is thechange in Nth

ij when the kth input is changed by σk (i.e. by 1 standard deviation).Since the uncertainties in the inputs are not very large, we only consider changesin Nth

ij that are linear in ξk. Thus we have the modified χ2,

χ2(ξk) =∑i,j

[Nth

ij (std) +∑npull

k=1 ckij ξk −Nex

ij

]2Nex

ij

+

npull∑k=1

ξ2k , (13)

where the additional ξ2k-dependent term is the penalty imposed for moving the value

of the kth input away from its standard value by σk ξk. The χ2 with pulls, whichincludes the effects of all theoretical and systematic uncertainties, is obtained byminimizing χ2(ξk) with respect to all the pulls ξk:

χ2pull = Minξk

[χ2(ξk)

]. (14)

In the calculation of χ2pull, we consider uncertainties in the flux, cross sections

etc. (as in Table 6), keeping the values of the oscillation parameters fixed whilecalculating Nex

ij and Nthij . However, in general, the values of the mass-squared differ-

ence ∆m231 and the mixing angles θ23 and θ13 can vary over a range corresponding

to the actual measurements of these parameters. Holding them fixed at particular


22

Table 6. Uncertainties for various quantities.18

Quantity ValueFlux normalization uncertainty 20%

Zenith angle dependence uncertainty 5%Cross section uncertainty 10%

Overall systematic uncertainty 5%

Tilt uncertainty Φδ ≈ Φ0(E)[1 + δ log

(EE0

)]with E0 = 2 GeV, σδ = 5% (see, e.g.,21)

values is equivalent to knowing the parameters to infinite precision, which is obvi-ously unrealistic. To take into account the uncertainties in the actual measurementof the oscillation parameters, we define the marginalized χ2 as18

χ2min = Min

[χ2(ξk) +

(|∆m2

31|true − |∆m231|

σ(|∆m231|)

)2

+

(sin2 2θtrue

23 − sin2 2θ23

σ(sin2 2θ23)

)2

+

(sin2 2θtrue

13 − sin2 2θ13

σ(sin2 2θ13)

)2]. (15)

The three terms added to χ2(ξk) are known as priors. Now, for our χ2 analysis, weproceed as follows, e.g. for the case of the mass hierarchy.

• Our aim is to see at what statistical significance the wrong hierarchy canbe ruled out. Our procedure gives the median sensitivity of the experimentin the frequentist approach.22

• We simulate the number of events in 10 bins in the measured energy Em

and 18 bins in the measured zenith angle cos θm for a set of true values forthe six neutrino parameters: θ12, θ23, θ13, ∆m2

21, ∆m231, δCP , and for a true

hierarchy. The true values are the current best fit values of the oscillationparameters and the true value of δCP is assumed to be zero. This is ourexperimental data – Nex

ij . Now we calculate the theoretical expectation ineach bin – Nth

ij assuming the wrong hierarchy, and calculate the χ2 betweenthese two datasets.• For the marginalization procedure, we allow θ23, θ23, |∆m2

31| and δCP tovary within the following ranges:θ23 ∈ [36, 54],θ13 ∈ [5.5, 11],|∆m2

31| ∈ [2.19, 2.62]× 10−3 eV2,δCP ∈ [−π, π].• In computing χ2

min, we add the priors for the neutrino parameters whichassigns a penalty for moving away from the true value. During marginal-ization, as the value of an oscillation parameters shifts further from its truevalue, Eq. (15) suggests that the corresponding prior will be larger resulting


23

νμ fluxνμ fluxνe fluxνe flux

Flux

( ν/

GeV

/ m

2 / PO

T [

× 10

-12 ] )

0.01

0.1

1

10

100

E [GeV]0 1 2 3 4 5 6 7 8 9 10

νμ CCνμ CC

σ cc/

E [

cm2 /

1038

Gev

]

0

0.2

0.4

0.6

0.8

log10E

−1 0 1 2 3

Fig. 16. The neutrino and antineutrino fluxes are shown in the left panel. The right panel showsthe ν and ν charged current cross sections.

in a higher χ2 value.• Finally, after adding the priors, we determine χ2

pull (see Eq. 14). Thisis a multi-dimensional parameter space minimization of the functionχ2(α, β, . . . ), where α, β, . . . are the parameters over which marginaliza-tion is required. For the purpose of this multi-minimization, we have usedthe NLopt library.23 We do the minimization first over the entire multi-dimensional parameter space to locate the global minimum approximately,and then use the parameters corresponding to this as a guess to carry outa local minimum search to locate the minimized χ2 within the parameterspace accurately. We carry out this minimization routine using a simplexalgorithm described in Ref.24 , and implemented within the NLopt library.

8. DUNE fluxes and atmospheric neutrino events

The fluxes and charged current cross sections used in our analysis are shown inFig. 16. These are similar to those used by the DUNE collaboration.

In Fig. 17, we show the number of νµ and νe atmospheric events with and withoutoscillations for an exposure of 350 kt-yr.


24

νμ oscillatedνμ oscillatedνμ unoscillatedνμ unoscillated

No.

of

even

ts

0

200

400

600

800

1000

1200

1400

1600

1800

2000

E [GeV]1 2 3 4 5 6 7 8 9 10

νe oscillatedνe oscillatedνe unoscillatedνe unoscillated

No.

of

even

ts

0

200

400

600

800

1000

1200

1400

1600

1800

2000

E [GeV]1 2 3 4 5 6 7 8 9 10

Fig. 17. νµ (left panel) and νe (right panel) atmospheric events for a 350 kt-yr LAr FD.

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